The discussion of some parts of Einstein's theory of gravitation[1] may perhaps gain in simplicity and clearness, if we base it on a principle similar to that of Hamilton, so much so indeed that Hamilton's name may properly be connected with it. Now that we are in possession of Einstein's theory we can easily find how this variation principle must be formulated for systems of different nature and also for the gravitation field itself.

Motion of a material point.

§ 1. Let a material point move under the influence of a force with the components. Let us vary every position occurring in the real motion in the way defined by the infinitely small quantities . If, in the varied motion, the position is reached at the same time as the position in the real motion, we shall have the equation

(1)

being the Lagrangian function and the integrals being taken over an arbitrary interval of time, at the beginning and the end of which the variations of the coordinates are zero. is supposed to be a force acting on the material point beside the forces that are included in the Lagrangian function.

§ 2. We may also suppose the lime / to he varied, so that in the varied motion the position ,/ is reached at the time . In the first term of (1) this does not make any difference if we suppose that for the extreme positions also . As to the second term we remark that the coordinates in the varied motion at the time may now be taken to be , , , if are the velocities in the real motion. In the second term we must therefore replace by , , . In the equation thus found we shall write for For the sake of uniformity we shall add to the three velocity components a fourth, which, however, necessarily must have the value 1 as we take for it . We shall also add to the three components of the force a fourth component, which we define as

(2)

and which therefore represents the work of the force per unit of time with the negative sign. Then we have instead of (1)

§ 3. In Einstein's theory the gravitation field is determined by certain characteristic quantities , functions of , among which there are 10 different ones, as

(5)

A point of fundamental importance is the connection between these quantities and the corresponding coefficients , with which we are concerned, when by an arbitrary substitution are changed for other coordinates . This connection is defined by the condition that

or shorter

be an invariant.

Putting

(6)

we find

(7)

Instead of (6) we shall also write

so that the set of quantities may be called the inverse of the set . Similarly, we introduce a set of quantities , the inverse of the set [3].

We remark here that in virtue of (5) and (7) and that likewise .

Our formulae will also contain the determinant of the quantities , which we shall denote by , and the determinant of the coefficients (absolute value: ). The determinant is always negative.

We may now, as has been shown by Einstein, deduce the motion of a material point in a gravitation field from the principle expressed by (3) if we take for the Lagrangian function

(8)

Motion of a system of incoherent material points.

§ 4. Let us now, following Einstein, consider a very large number of material points wholly free from each other, which are moving in a gravitation field in such a way that at a definite moment the velocity components of these points are continuous functions of the coordinates. By taking the number very large we may pass to the limiting case of a continuously distributed matter without internal forces.

Evidently the laws of motion for a system of this kind follow immediately from those for a single material point. If is the density and an clement of volume we may write instead of (8)

(9)

If now we wish lo extend equation (3) to the whole system we must multiply (9) by and integrate with respect to and .

In the last term of (3) we shall do so likewise after having replaced the components by , so that in what follows will represent the external force per unit of volume.

If further we replace by , an element of the four-dimensional extension , and put

(10)

(11)

we find the following form of the fundamental theorem.

Let a variation of the motion of the system of material points be defined by the infinitely small quantities , which are arbitrary continuous functions of the coordinates within an arbitrarily chosen finite space , at the limits of which they vanish. Then we have, if the integrals are taken over the space , and the quantities are left unchanged,

(12)

For the first term we may write

if denotes the change of at a fixed point of the space .

The quantity and therefore also the integral is invariant when we pass to another system of coordinates.[4]

§ 5. The equations of motion may be derived from (12) in the following way. When the variations have been chosen, the varied motion of the matter is perfectly defined, so that the changes of the density and of the velocity components are also known. For the variations at a fixed point of the space we find

(13)

where

(14)

(Therefore: ).

If for shortness we put

(15)

so that , and

(16)

we have

so that, with regard to (14),

(17)

If after multiplication by this expression is integrated over the space the first term on the right hand side vanishes, being at the limits. In the last two terms only the variations occur, but not their differential coefficients, so that according to our fundamental theorem, when these terms are taken together, the coefficient of each must vanish. This gives the equations of motion[5]

(18)

which evidently agree with (4), or what comes to the same, with

(19)

In virtue of (18) the general equation (17), which holds for arbitrary variations that need not vanish at the limits of , becomes

(20)

§ 6. We can derive from this the equations for the momenta and the energy.

Let us suppose that only one of the four variations differs from 0 and let this one, say , have a constant value. Then (14) shows that for each value of that is not equal to

(21)

while all 's without an index vanish.

Putting first and then , and replacing at the same time in the latter case by , we find for the right hand side of (20)

The material system together with its state of motion has been shifted in the direction of the coordinate over a distance . If the gravitation field had participated in this shift, would have been equal to . As, however, the gravitation field has been left unchanged, in this last expression must be diminished by , the index meaning that we must keep constant the quantities and consider only the variability of the coefficients . Hence

Substituting this in (22) and putting

(23)

we find

(24)

The first three of these equations ( = 1, 2,3) refer to the momenta; the fourth ( = 4) is the equation of energy. As we know already the meaning of we can easily see that of the other quantities. The stresses are ; the components of the momentum per unit of volume ; the components of the flow of energy . Further is the energy per unit of volume. The quantities

are the momenta which the gravitation field imparts to the material system per unit of time and unit of volume, while the energy which the system draws from that field is given by .

An electromagnetic system In the gravitation field.

§ 7. We shall now consider charges moving under the influence of external forces in a gravitation field; we shall determine the motion of these charges and the electromagnetic field belonging to them. The density of the charge will be supposed to be a continuous function of the coordinates; the force per unit of volume will be denoted by and the velocity of the point of a charge by . Further we shall again introduce the notation (10).

In Einstein's theory the electromagnetic field is determined by two sets, each of four equations, corresponding to well known equations in the theory of electrons. We shall consider one of these sets as the mathematical description of the system to which we have to apply Hamilton's principle; the second set will be found by means of this application.

The first set, the fundamental equations, may be written in the form

(25)

the quantities [7] on the left hand side being subject to the conditions

(26)

so that they represent 6 mutually independent numerical values. These are the components of the electric force and the magnetic force . We have indeed

(27)

and it is thus seen that the first three of the formulae (25) express the connection between the magnetic field and the electric current. The fourth shows how the electric field is connected with the charge.

On passing to another system of coordinates we have for the transformation formula

which can easily be deduced, while for we shall assume the formula

(28)

In virtue of this assumption the equations (25) are covariant for any change of coordinates.

§ 8. Beside we shall introduce certain other quantities which we define by

(29)

or with regard to (26)

(30)

in which last equation the bar over means that in the sum each combination of two numbers occurs only once.

According to (30) these latter are homogeneous and linear functions of the former and as (because of (5)) the coefficient of in is equal to the coefficient of in , there exists a homogeneous quadratic function of , which, when differentiated with respect to these quantities, gives . Therefore

(34)

and

(35)

If, as in (34), we have to consider derivatives of , this quantity will be regarded as a quadratic function of the quantities .

The quantity can now play the same part as the quantity that is represented by the same letter in §§ 4 — 6. Again is invariant when the coordinates are changed.[10]

§ 10. We shall define a varied motion of the electric charges by the quantities and we shall also vary the quantities , so far as can be done without violating the connections (25) and (26). The possible variations may be expressed in and four other infinitesimal quantities which we shall presently introduce. Our condition will be that equation (12) shall be true if, leaving the gravitation field unchanged, we take for and any continuous functions of the coordinates which vanish at the limits of the domain of integration. We shall understand by , , the variations at a fixed point of this space. The variations are again determined by (13) and (14), and we have, in virtue of (26) and (25),

If therefore we put

(36)

we must have

It can be shown that quantities satisfying these conditions may be expressed in terms of four quantities by means of the formulae

(37)

Here and are the numbers that remain when of 1, 2, 3, 4 we omit and , the choice of the value of and that of being such that the order can be derived from the order 1, 2, 3, 4 by an even number of permutations each of two numbers.

§ 11. By (31), (36) and (37) we have

(38)

Here, in the transformation of the first term on the right hand side it is found convenient to introduce a new notation for the quantities . We shall put

If we multiply (40) by and integrate over the space the first term on the right hand side vanishes. Therefore (12) requires that in the subsequent terms the coefficient of each and of each be 0. Therefore

(41)

and

(42)

by which (40) becomes

(43)

In (41) we have the second set of four electromagnetic equations, while (42) determines the forces exerted by the field on the charges. We see that (42) agrees with (19) (namely in virtue of (31)).

§ 12. To deduce also the equations for the momenta and the energy we proceed as in § 6. Leaving the gravitation field unchanged we shift the electromagnetic field, i. e. the values of and in the direction of one of the coordinates, say of , over a distance defined by the constant variation so that we have

From (36), (14) and (37) we can infer what values must then be given to the quantities . We must put and for [12]

For we must substitute the expression (cf. § 6)

where the index attached to the second derivative indicates that only the variability of the coefficients (depending on ) in the quadratic function must be taken into consideration. The equation for the component which we finally find from (43) may be written in the form

(44)

where

(45)

and for

(46)

Equations (44) correspond exactly to (24). The quantities have the same meaning as in these latter formulae and the influence of gravitation is determined by in the same way as it was formerly by .

We may remark here that the sum in (45) consists of three and that in (46) (on account of (39)) of two terms.

Referring to (35), we find f.i. from (45)

while (46) gives

The differential equations of the gravitation field.

§ 13. The equations which, for a given material or electromagnetic system, determine the gravitation field caused by it can also be derived from a variation principle. Einstein has prepared the way for this in his hist paper by introducing a quantity which he calls and which is a function of the quantities and their derivatives, without further containing anything that is connected with the material or the electromagnetic system. All we have to do now is to add to the left hand side of equation (12) a term depending on that quantity . We shall write for it the variation of

where is a universal constant, while is what Einstein calls , with the same or the opposite sign[13]. We shall now require that

(47)

not only for the variations considered above but also for variations of the gravitation field defined by , if these too vanish at the limits of the field of integration.

To obtain now

we have to add to the right hand side of (17) or (40), first the change of caused by the variation of the quantities , viz.

and secondly the change of multiplied by . This latter change is

where bas been written for the derivative .

As

we may replace the last term by

§ 14. As we have to proceed now in the same way in the case of a material and in that of an electromagnetic system we need consider only the latter. The conclusions drawn in § 11 evidently remain valid, so that we may start from the equation which we obtain by adding the new terms to (43). We therefore have

(48)

When we integrate over , the first two terms on the right hand side vanish. In the terms following them the coefficient of each must be 0, so that we find

(49)

These are the differential equations we sought for. At the same time (48) becomes

(50)

§ 15. Finally we can derive from this the equations for the momenta and the energy of the gravitation field. For this purpose we impart a virtual displacement to this field only (comp. §§ 6 and 12). Thus we put and

Evidently

and (comp. § 12)

After having substituted these values in equation (50) we can deduce from it the value of .

If we put

(51)

and for

(52)

the result takes the following form

(53)

Remembering what has been said in § 12 about the meaning of , we may now conclude that the quantities have the same meaning for the gravitation field as the quantities for the electro-magnetic field (stresses, momenta etc.). The index denotes that belongs to the gravitation field.

If we add to (53) the equations (44), after having replaced in them by , we obtain

(54)

where

The quantities represent the total stresses etc. existing in the system, and equations (54) show that in the absence of external forces the resulting momentum and the total energy will remain constant.

We could have found directly equations (54) by applying formula (50) to the case of a common virtual displacement imparted both to the electromagnetic system and to the gravitation field.

Finally the differential equations of the gravitation field and the formulae derived from them will be quite conform to those given by Einstein, if in we substitute for the function he has chosen.

§ 16. The equations that have been deduced here, though mostly of a different form, correspond to those of Einstein. As to the covariancy, it exists in the case of equations (18), (24), (41), (42) and (44) for any change of coordinates. We can be sure of it because is an invariant.

On the contrary the formulae (49), (53) and (54) have this property only when we confine ourselves to the systems of coordinates adapted to the gravitation field, which Einstein has recently considered. For these the covariancy of the formulae in question is a consequence of the invariancy of which Einstein has proved by an ingenious mode of reasoning.

↑In these formulae we have put between parentheses behind the sign of summation the index with respect to which the summation must be effected, which means that the values 1, 2, 3, 4 have to be given to it successively. In the same way two or more indices behind the sign of summation will indicate that in the expression under this sign these values have to be given to each of the indices. f. i. means that each of the four values of a has to be combined with each of the four values of .

↑The circumstance that (21) does not hold for might lead us to exclude this value from the two sums. We need not, however, introduce this restriction, as the two terms that are now written down too much, annul each other.

↑The quantities are connected with the components of the tensor introduced by Einstein by the equations .

Substituting for an expression similar to (29) with other letters as indices, we have

The last two steps of this transformation, which rest on and , will need no further explanation. In a similar way we may proceed (comp. the following notes) in many other cases, using also the relations and (the latter for ), which are similar to and .

↑If we start from the equation for that corresponds to (29) and if we use (7) and (28), attending to , we find

This may be transformed in two steps (comp. the preceding note) to

In this way we may proceed further, after first expressing as a function of by means of (32).

↑Instead of (35) we may write and now we have according to (28) and (33)

while

↑The quantities are connected with the quantities introduced by Einstein by the equation .

↑I have not yet made out which sign must be taken to get a perfect conformity to Einstein's formulae.

This work is in the public domain in the United States because it was published before January 1, 1923.

The author died in 1928, so this work is also in the public domain in countries and areas where the copyright term is the author's life plus 80 years or less. This work may also be in the public domain in countries and areas with longer native copyright terms that apply the rule of the shorter term to foreign works.